In the UK there is a non-trivial relationship between the two.
On the west coast (in Seattle specifically -- but I believe the same holds true in LA or the bay area) nearly all money is new money, which forces social mobility to be tightly coupled with economic mobility.
Another significant factor is that, culturally, Americans (in my experience) more closely associate wealth with success than any other nation and success correlates directly to social value (and mobility).
While it might be possible to move up the economic rungs without moving up socially, it's not generally possible to move up the social ladder without correspondingly increasing your wealth.
(Disclaimer: I'm not by any means a member of the social elite -- to the extent that such a thing exists in Seattle -- nor am I by any means "rich")
The article mystifies me from a global perspective. Americans to me seem obsessed with upward mobility. The "American Dream" is a dream of upward mobility, for goodness sake.
Why do we resent taxes on the rich, even when we're poor? Because we expect one day that we might be that rich person. One's skin color or religion is largely irrelevant in the US compared with the amount of money that you have. This is not the truth in many places.